Intersecting sets in probability spaces and Shelah's classification

TL;DR

This paper explores complex intersection patterns of events in probability spaces, their connections to model-theoretic stability, and their implications for higher arity generalizations of de Finetti's theorem, Ramsey theory, and hypergraph regularity.

Contribution

It surveys recent results on intricate event patterns indexed by multiple parameters and links these to advanced topics in probability, combinatorics, and model theory.

Findings

Certain intersection patterns cannot occur for large event collections

Connections established between probability event patterns and model-theoretic stability

Results relate to higher arity de Finetti theorems and hypergraph regularity

Abstract

For $n \in \mathbb{N}$ and $\varepsilon > 0$, given a sufficiently long sequence of events in a probability space all of measure at least $\varepsilon$, some $n$ of them will have a common intersection. A more subtle pattern: for any $0 < p < q < 1$, we cannot find events $A_i$ and $B_i$ so that $\mu \left( A_i \cap B_j \right) \leq p$ and $\mu \left( A_j \cap B_i\right) \geq q$ for all $1 < i < j < n$, assuming $n$ is sufficiently large. This is closely connected to model-theoretic stability of probability algebras. We survey some results from our recent work on more complicated patterns that arise when our events are indexed by multiple indices. In particular, how such results are connected to higher arity generalizations of de Finetti's theorem in probability, structural Ramsey theory, hypergraph regularity in combinatorics, and model theory.